jan 29 WaitingTimes_R2

# jan 29 WaitingTimes_R2 - probability of S on the n th trial...

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C.D. Cutler STAT 333 Discrete Waiting Time Random Variables Let E be a possible event of a stochastic process X 1 , X 2 , X 3 , . . . . Let X = waiting time (number of steps or trials) until E Frst occurs in the sequence. The potential range of X is { 1 , 2 , 3 , 4 , . . . } ∪ {∞} where X = n means E Frst occurs on trial n and X = means that E never occurs. ±or each x in the potential range we deFne f ( x ) = P ( X = x ). This is the probability mass function (p.m.f.) of X . We must include the value f ( ) = P ( E never occurs). X is called proper if f ( ) = 0 and improper if f ( ) > 0. Proper random variables are those with which you are familiar from other courses; they satisfy x =1 f ( x ) = 1 (there is no probability at and their range can be restricted to the integers). Proper random variables can be further classiFed into two types: Short proper waiting times: E ( X ) < Null (long) waiting times: E ( X ) = (*) In the special case that we have a sequence of independent S or ± trials, where the
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Unformatted text preview: probability of S on the n th trial is p n , we can determine whether or not the waiting time variable “ X = number of trials until the Frst S” is proper or not by applying the Sum-Product Lemma below. (The Sum-Product Lemma is a mathematical result from real analysis.) The Sum-Product Lemma: Suppose 0 < p n < 1 for each n . Then ∞ p n =1 (1-p n ) = c > if and only if ∞ s n =1 p n < ∞ application: in the special case (*) noted above, we have P ( X > n ) = P ( FF . . . F ) = (1-p 1 )(1-p 2 ) ··· (1-p n ) . Therefore f ( ∞ ) = lim n →∞ (1-p 1 )(1-p 2 ) ··· (1-p n ) = P ∞ n =1 (1-p n ), and so the Sum-Product Lemma can be applied to determine whether f ( ∞ ) > 0 or f ( ∞ ) = 0....
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